CSL202: Discrete Mathematical Structures

Ragesh Jaiswal, CSE, IIT Delhi

How do we design a good hash function?

A set S of keys from a universeU ={0,1, ...,m−1} is supposed to be stored in a table of sizen with indices T ={0,1, ...,n−1}.

Assume collisions are resolved using auxiliary data structure.

What we need is a hash functionh :U →T with the following main requirements:

1 The hash function should minimize the number of collisions.

2 The space used should be proportional to the number of keys stored. (i.e.,n≈ |S|)

How do we design a good hash function?

A set S of keys from a universeU ={0,1, ...,m−1} is supposed to be stored in a table of sizen with indices T ={0,1, ...,n−1}.

Assume collisions are resolved using auxiliary data structure.

What we need is a hash functionh :U →T with the following main requirements:

1 The hash function should minimize the number of collisions.

2 The space used should be proportional to the number of keys stored. (i.e.,n≈ |S|)

Claim 1: Ifm>n, then for anyh there exists a key setS such that h has collision w.r.t. S (i.e., ∃x,y ∈S,h(x) =h(y))

How do we design a good hash function?

A set S of keys from a universeU ={0,1, ...,m−1} is supposed to be stored in a table of sizen with indices T ={0,1, ...,n−1}.

Assume collisions are resolved using auxiliary data structure.

What we need is a hash functionh :U →T with the following main requirements:

1 The hash function should minimize the number of collisions.

2 The space used should be proportional to the number of keys stored. (i.e.,n≈ |S|)

Claim 1: Ifm>n, then for anyh there exists a key setS such that h has collision w.r.t. S (i.e., ∃x,y ∈S,h(x) =h(y))

Claim 1.1: Any fixed hash functionh:U →T, must map at leastd^m_neelements of U to some index in the setT.

How do we design a good hash function?

A set S of keys from a universeU ={0,1, ...,m−1} is supposed to be stored in a table of sizen with indices T ={0,1, ...,n−1}.

Assume collisions are resolved using auxiliary data structure.

What we need is a hash functionh :U →T with the following main requirements:

1 The hash function should minimize the number of collisions.

2 The space used should be proportional to the number of keys stored. (i.e.,n≈ |S|)

Claim 1: Ifm>n, then for anyh there exists a key setS such that h has collision w.r.t. S (i.e., ∃x,y ∈S,h(x) =h(y)) Claim 2: For any fixed key set S such that |S| ≤n, there exists a hash function such thath has no collisions w.r.t. S.

How do we design a good hash function?

A set S of keys from a universeU ={0,1, ...,m−1} is supposed to be stored in a table of sizen with indices T ={0,1, ...,n−1}.

Collisions are resolved using auxiliary data structure.

What we need is a hash functionh :U →T with the following main requirements:

1 The hash function should minimize the number of collisions.

2 The space used should be proportional to the number of keys stored. (i.e.,n≈ |S|)

Claim 1: Ifm>n, then for anyh there exists a key setS such that h has collision w.r.t. S (i.e., ∃x,y ∈S,h(x) =h(y)) Claim 2: For any fixed key set S such that |S| ≤n, there exists a hash function such thath has no collisions w.r.t. S. The issue is that the key setS is not known a-priori. That is, before using the data structure.

Question: How do we solve this problem then?

How do we design a good hash function?

A setS of keys from a universeU={0,1, ...,m−1}is supposed to be stored in a table of sizenwith indicesT ={0,1, ...,n−1}.

Collisions are resolved using auxiliary data structure.

What we need is a hash function h:U→T with the following main requirements:

1 The hash function should minimize the number of collisions.

2 The space used should be proportional to the number of keys stored. (i.e.,n≈ |S|)

Claim 1: If m>n, then for anyhthere exists a key set S such thath has collision w.r.t. S (i.e., ∃x,y ∈S,h(x) =h(y))

Claim 2: For any fixed key setS such that|S| ≤n, there exists a hash function such thath has no collisions w.r.t. S.

The issue is that the key setS is not known a-priori. That is, before using the data structure.

Question: How do we solve this problem then?

Randomlyselect a hash function from afamilyHof hash functions.

How do we design a good hash function?

A setSof keys from a universeU={0,1, ...,m−1}is supposed to be stored in a table of sizenwith indicesT={0,1, ...,n−1}.

Collisions are resolved using auxiliary data structure.

What we need is a hash functionh:U→T with the following main requirements:

1 The hash function should minimize the number of collisions.

2 The space used should be proportional to the number of keys stored. (i.e.,n≈ |S|)

The issue is that the key set Sis not known a-priori. That is, before using the data structure.

Question: How do we solve this problem then?

Randomlyselect a hash function from afamilyHof hash functions.

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤1 n.

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤ 1 n.

Theorem: Consider hashing using a 2-universal hash function family. Considertinsert operations, the expected cost of each operation is at most (1 +t/n).

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤ 1 n.

Theorem: Consider hashing using a 2-universal hash function family. Considertinsert operations, the expected cost of each operation is at most (1 +t/n).

Proof sketch: Consider any keyx. The expected number of keys in locationh(x) is at most t/n.

Question: Can you think of a 2-universal hash function family?

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤ 1 n.

Theorem: Consider hashing using a 2-universal hash function family. Considertinsert operations, the expected cost of each operation is at most (1 +t/n).

Proof sketch: Consider any keyx. The expected number of keys in locationh(x) is at most t/n.

Question: Can you think of a 2-universal hash function family?

Simple answer: The set ofallfunctions fromU toT.

Do you see any issues with using this hash function family?

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤ 1 n.

Theorem: Consider hashing using a 2-universal hash function family. Considertinsert operations, the expected cost of each operation is at most (1 +t/n).

Proof sketch: Consider any keyx. The expected number of keys in locationh(x) is at most t/n.

Question: Can you think of a 2-universal hash function family?

Simple answer: The set ofallfunctions fromU toT.

Do you see any issues with using this hash function family? The description of any hash function from this family is large.

Question: Can we design a morecompacthash function family?

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤ 1 n.

Theorem: Consider hashing using a 2-universal hash function family. Considertinsert operations, the expected cost of each operation is at most (1 +t/n).

A compact 2-universal hash function family:

Letm≤p≤2m.

H={ha,b|a∈ {1, ...,p−1},b∈ {0, ...,p−1}}and h_a,b(x) = ((ax+b) modp) modn.

How many functions doesH have?

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤ 1 n.

Theorem: Consider hashing using a 2-universal hash function family. Considertinsert operations, the expected cost of each operation is at most (1 +t/n).

A compact 2-universal hash function family:

Letm≤p≤2m.

H={ha,b|a∈ {1, ...,p−1},b∈ {0, ...,p−1}}and h_a,b(x) = ((ax+b) modp) modn.

How many functions doesH have? p(p−1)

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤ 1 n.

Theorem: Consider hashing using a 2-universal hash function family. Considertinsert operations, the expected cost of each operation is at most (1 +t/n).

A compact 2-universal hash function family:

Letm≤p≤2m.

H={ha,b|a∈ {1, ...,p−1},b∈ {0, ...,p−1}}and h_a,b(x) = ((ax+b) modp) modn.

How many functions doesH have? p(p−1) Theorem: H is 2-universal.

Definition (2-universality)

A hash function familyH is said to be 2-universal iff:

∀x,y ∈U,x 6=y,Pr_h←H[h(x) =h(y)]≤ 1 n.

Theorem: Consider hashing using a 2-universal hash function family. Considertinsert operations, the expected cost of each operation is at most (1 +t/n).

A compact 2-universal hash function family:

Letm≤p≤2m.

H={ha,b|a∈ {1, ...,p−1},b∈ {0, ...,p−1}}and h_a,b(x) = ((ax+b) modp) modn.

Theorem: H is 2-universal.

Theorem: H is 2-universal.

Proof sketch

Letg_a,b(x) = (ax+b) modp. So,h_a,b(x) =g_a,b(x) modn.

Consider anyx,y∈ {0, ...,p−1}such thatx6=y.

Claim 1: Ifh_a,b(x) =h_a,b(y), theng_a,b(x) =g_a,b(y) modn.

Theorem: H is 2-universal.

Proof sketch

Letg_a,b(x) = (ax+b) modp. So,h_a,b(x) =g_a,b(x) modn.

Consider anyx,y∈ {0, ...,p−1}such thatx6=y.

Claim 1: Ifh_a,b(x) =h_a,b(y), theng_a,b(x) =g_a,b(y) modn.

Claim 2: For allα, β∈ {0, ...,p−1}:

Pr[g_a,b(x) =αandg_a,b(y) =β] =

(0 ifα=β

1

p(p−1) otherwise

Theorem: H is 2-universal.

Proof sketch

Letg_a,b(x) = (ax+b) modp. So,h_a,b(x) =g_a,b(x) modn.

Consider anyx,y∈ {0, ...,p−1}such thatx6=y.

Claim 1: Ifh_a,b(x) =h_a,b(y), theng_a,b(x) =g_a,b(y) modn.

Claim 2: For allα, β∈ {0, ...,p−1}:

Pr[g_a,b(x) =αandg_a,b(y) =β] =

(0 ifα=β

1

p(p−1) otherwise Claim 3: We have:

Pr[h_a,b(x) =h_a,b(y)] = |{(α, β) :α6=β andα≡β modn}|

p(p−1) ≤ 1

n.